Module 3 : Differentiation and Mean Value Theorems
Lecture 8 : Chain Rule [Section 8.1]
Since, both and are differentiable everywhere, by chain rule and theorem 7.1.8, is also is differentiable at every in and
8.1.3
Example (Differentiation of rational powers) :
Let , be any positive differentiable function and , Then the function is differentiable with
To see this, let us consider the case when . Let , where are both positive integers .
Let . Then
where Note that Thus, by chain rule
and 
are differentiable everywhere, by chain rule and theorem 7.1.8, 
is also is differentiable at every 
in 
and 
, be any positive differentiable function and 
, Then the function 
is differentiable with 
. Let 
, where 
are both positive integers . 
. Then
Note that 
Thus, by chain rule
